Calogero type bounds in two dimensions

Abstract

For a Schr\"odinger operator on the plane $\mathbb{R}^2$ with electric potential $V$ and Aharonov--Bohm magnetic field we obtain an upper bound on the number of its negative eigenvalues in terms of the $L^1(\mathbb{R}^2)$-norm of $V$. Similar to Calogero's bound in one dimension, the result is true under monotonicity assumptions on $V$. Our proof method relies on a generalisation of Calogero's bound to operator-valued potentials. We also establish a similar bound for the Schr\"odinger operator (without magnetic field) on the half-plane when a Dirchlet boundary condition is imposed and on the whole plane when restricted to antisymmetric functions.